How linear surface waves are affected by a current with constant vorticity

The interaction of surface waves with Couette-type current with uniform vorticity is a well suited problem for students approaching the theory of surface waves. The problem, although mathematically simple, contains rich physics, and is moreover important in several situations from oceanography and marine technology to microfluidics. We here lay out a simple two-dimensional theory of waves propagating upon a basic flow of uniform vorticity of constant depth. The dispersion relation is found, showing how the shearing current introduces different phase velocities for upstream and downstream propagating waves. The role of surface tension is discussed and applied to the case of a wave pattern created by a moving source, stationary as seen by the source. We conclude by discussing how the average potential and kinetic energies are no longer equal in the presence of shear.